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## Annales Polonici Mathematici

2012 | 106 | 1 | 31-40
Tytuł artykułu

### Markov's property for kth derivative

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EN
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EN
Consider the normed space $(ℙ(ℂ^{N}),||·||)$ of all polynomials of N complex variables, where || || a norm is such that the mapping $L_{g}: (ℙ(ℂ^{N}),||·||) ∋ f ↦ gf ∈ (ℙ(ℂ^{N}),||·||)$ is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality
$|∂/∂z_{j} P|| ≤ M(deg P)^{m} ||P||$, j = 1,...,N, $P ∈ ℙ(ℂ^{N})$,
with positive constants M and m is equivalent to the inequality
$||∂^{N}/∂z₁...∂z_{N} P|| ≤ M'(deg P)^{m'}||P||$, $P ∈ ℙ(ℂ^{N})$,
with some positive constants M' and m'. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras. In addition, we give a nontrivial example of Markov's inequality in the Wiener algebra of absolutely convergent trigonometric series and show that the Banach algebra approach to Markov's property furnishes new tools in the study of polynomial inequalities.
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Rocznik
Tom
Numer
Strony
31-40
Opis fizyczny
Daty
wydano
2012
Twórcy
autor
• Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
autor
• Institute of Mathematical and Natural Sciences, State Higher Vocational School in Tarnów, Mickiewicza 8, 33-100 Tarnów, Poland
autor
• Institute of Mathematical and Natural Sciences, State Higher Vocational School in Tarnów, Mickiewicza 8, 33-100 Tarnów, Poland
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